Capillary Fracturing in Granular Media

We study the displacement of immiscible fluids in deformable, noncohesive granular media. Experimentally, we inject air into a thin bed of water-saturated glass beads and observe the invasion morphology. The control parameters are the injection rate, the bead size, and the confining stress. We identify three invasion regimes: capillary fingering, viscous fingering, and “capillary fracturing,” where capillary forces overcome frictional resistance and induce the opening of conduits. We derive two dimensionless numbers that govern the transition among the different regimes: a modified capillary number and a fracturing number. The experiments and analysis predict the emergence of fracturing in fine-grained media under low confining stress, a phenomenon that likely plays a fundamental role in many natural processes such as primary oil migration, methane venting from lake sediments, and the formation of desiccation cracks.

Figure 1

Experimental setup: a thin bed (thickness $b$) of water-saturated glass beads (mean diameter $d$) is confined in a cylindrical acrylic cell (internal diameter $L$). Vertical confinement is supplied by a weight, $w$, placed on a disk (“lid”) that rests on top of the beads. The disk is slightly smaller than the cell to allow fluids (but not particles) to leave the cell. Air is injected into the center of the cell at a fixed flow rate $q$. Time-lapse images are captured by a camera placed underneath the cell.

Figure 2

Examples of experimentally observed patterns. We classify these patterns into three regimes: viscous fingering, capillary fingering, and fracturing. In these difference images, obtained by subtracting the initial image prior to air invasion, air shows as clear white, water and undisturbed beads as black, and deformed regions as a sparkly halo (bead displacements change the reflected light). Experimental conditions: $d=360\mu \mathrm{m}$; $w=181\mathrm{N}$ (a),(b) and 3 N (c); and $q=100$ (a), 0.1 (b), and $1\mathrm{mL}/\min$ (c).

Figure 3

Phase diagrams of fluid-fluid displacement patterns in the experiments. The diagrams demonstrate how changing the control parameters affects the patterns. Diagram (a) shows that the transition from viscous fingering (VF) to capillary fingering (CF) occurs at $q/b\sim d^2$ (fixing $w=181\mathrm{N}$). Intermediate patterns are marked as VF/CF. Diagram (b) shows that the transition from capillary fingering to fracturing (FR) occurs at $w\sim d^{-1}$ ($q=0.1\mathrm{mL}/\min$). Diagram (c) shows that the transition from fingering to fracturing is independent of $q/b$ ($d=360\mu \mathrm{m}$).

Figure 4

Phase diagram of drainage in granular media, showing three invasion regimes: viscous fingering (VF), capillary fingering (CF), and fracturing (FR). The tendency to fracture is characterized by the “fracturing number” $N_f$: drainage is dominated by fracturing in systems with $N_f\gg 1$. At lower $N_f$ values, the type of fingering depends on the modified capillary number, ${\mathrm{Ca}}^{*}$.